Construction of Bernstein-based words and their patterns

With inspiration of the definition of Bernstein basis functions and their recurrence relation, in this paper we give construction of new concept so-called Bernstein-based words. By classifying these Bernstein-based words as first and second kind, we investigate their some fundamental properties involving periodicity and symmetricity. Providing schematic algorithms based on tree diagrams, we also illustrate the construction of the Bernstein-based words. Moreover, we give computational implementations of Bernstein-based words in the Wol-fram Language. By executing these implementations, we present some tables of Bernstein-based words and their decimal equivalents. In addition, we present black-white and 4-colored patterns arising from the Bernstein-based words with their potential applications. We also give some finite sums and generating functions for the lengths of the Bernstein-based words. We show that these functions are of relationships with the Catalan numbers, the centered m-gonal numbers, the Laguerre polynomials, certain finite sums, and hypergeometric functions. We also raise some open questions and provide some comments on our results. Finally, we investigate relations between the slopes of the Bernstein-based words and the Farey fractions.